Differential calculus for generalized geometry and geometric Lax flows

Abstract

Employing a class of generalized connections, we describe certain differential complices Ω * ( ) , ̃ constructed from ∧ *and study some of their basic properties, where = ⊕ * is the generalized tangent bundle on . A number of classical geometric notions are extended to , such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenböck identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest.